Matching-Induced Dependent Sampling

• Matching-Induced Dependent Sampling is a framework where matching and pairing processes alter the joint distribution of data, challenging conventional i.i.d. assumptions.
• The methods address mechanisms like informative survey sampling, contextual record linkage, and capacity constraints in matching markets to manage induced dependencies.
• Advanced correction techniques, such as pairwise pseudo-posteriors and matrix completion, mitigate bias and non-identifiability arising from the matching process.

Searching arXiv for the cited papers and topic to ground the article in current arXiv metadata. Attempting arXiv lookup for ([1710.10102](/papers/1710.10102)) and related matching-dependent-sampling papers. Matching-induced dependent sampling refers to sampling and observation regimes in which the act of forming matches, pairs, or constrained allocations changes the joint law of the observed data. The induced dependence may arise because units are selected jointly, because contextual matching links records that were originally independent, because sequential or permutation-based search rules alter the law of the observed match, or because capacity constraints determine which matrix entries can be observed in a matching market. Across these settings, the common statistical issue is that procedures justified under i.i.d. sampling or purely marginal weighting can become biased, non-identifiable, or inferentially invalid unless the dependence created by the matching mechanism is modeled explicitly [(Williams et al., 2017); (Carpentier et al., 2016); (Arratia et al., 2012); (O'Neill, 2021); (Duan et al., 30 Oct 2025)].

1. Mechanisms that induce dependence

The dependence structures studied under this theme are heterogeneous but structurally analogous: the sampling rule does not merely reveal data, it also couples the observations. In informative survey sampling, pairwise inclusion indicators can remain dependent even asymptotically, as in household or spouse sampling. In record linkage, contextual matching creates paired samples that are not random draws from the joint law of the variables of interest. In coincidence problems, the distribution of the observed match depends on how the match is searched for. In matching markets, observed entries arise from whole matchings subject to row and column constraints rather than from independent entrywise missingness [(Williams et al., 2017); (Carpentier et al., 2016); (Arratia et al., 2012); (O'Neill, 2021); (Duan et al., 30 Oct 2025)].

Setting	Dependence source	Statistical consequence
Informative survey sampling	Joint inclusion via $\pi_{ij}=P(\delta_i=1\cap \delta_j=1)$	Marginal pseudo posterior can be biased
Independent databases after matching	Pairing rule based on contextual variables $Z$	Matched sample is not an independent draw from $(X,Y)$
Coincidence and matching rules	Sequential memory or random permutation	Output law depends on the matching procedure
Two-sided matching markets	Capacity-constrained matching matrices $X_t$	Missingness pattern is structurally dependent

A central point is that dependence is often induced at the same level as the inferential target. In household and spouse designs, the target may be conditional behavior within sampled pairs; in matching markets, the target may be a linear functional of a latent reward matrix under a constrained observation process; in record linkage, the object of interest is a relationship learned from matched or stratified marginal samples rather than from genuine paired observations. This suggests that the relevant correction must usually act at the pair, match, or constrained-allocation level rather than only at the level of single marginal observations.

2. Pairwise inclusion and Bayesian pseudo-posterior correction

"Bayesian Pairwise Estimation Under Dependent Informative Sampling" formulates the clearest survey-sampling response to persistent pair dependence. The paper considers inference for a population model $p(y_i\mid \lambda)$ under informative sampling, where inclusion indicators $\delta_i$ are correlated with responses $y_i$. A standard correction uses marginal inclusion probabilities $\pi_i=P(\delta_i=1)$ to exponentiate each unit likelihood in a pseudo posterior, but that correction is valid only under designs in which pairwise inclusion dependence asymptotically attenuates to $0$. Household sampling, spouse sampling, and clustered multi-stage designs with fixed cluster size violate that restriction because dependence within the sampled cluster does not vanish (Williams et al., 2017).

The proposed alternative shifts from first-order to second-order inclusion information. Pairwise weights are built from

$$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$

and aggregated into

$Z$0

The pairwise pseudo posterior is rearranged as

$Z$1

Unnormalized weights $Z$2 are first converted to

$Z$3

and then normalized so that the final weights sum to the sample size $Z$4. The intent is to preserve the overall scale of the likelihood while correcting for pairwise selection pathways.

The main consistency result replaces the earlier asymptotic-independence requirement with conditions on higher-order inclusion probabilities. Under conditions (A1)–(A7), the pairwise pseudo posterior contracts to the truth $Z$5 in the pseudo-Hellinger metric: $Z$6 The specific design conditions include second-order probabilities bounded away from zero, bounded ratios of third- to second-order probabilities, asymptotic factorization of fourth-order probabilities, and a constant sampling fraction. The substantive consequence is that pairwise dependence may persist, provided higher-order selection structure is controlled. In the paper’s empirical and simulation evidence, pairwise weights reduced bias and mean squared error relative to marginal weights in conditional spouse or household subpopulation settings.

3. Matching as linkage, stratification, and distributional inversion

"Learning relationships between data obtained independently" studies a different but closely related form of dependence. Two variables of interest, $Z$7 and $Z$8, are observed in separate databases, with optional contextual variables $Z$9. Standard matching uses $(X,Y)$0 to create record pairs, but once such a pairing rule is imposed, the resulting sample $(X,Y)$1 is no longer an independent draw from the joint law of $(X,Y)$2. The paper therefore treats matching as a preprocessing device and learns the relationship from matched or context-specific marginal samples rather than from the matched pairs themselves (Carpentier et al., 2016).

The model is

$(X,Y)$3

with $(X,Y)$4 independent of $(X,Y)$5, and $(X,Y)$6 assumed monotone. The conditional density of $(X,Y)$7 satisfies

$(X,Y)$8

If contextual variables are available, the procedure first defines subsets $(X,Y)$9 and $X_t$0; if they are absent, the full samples are used. The central estimator is

$X_t$1

where $X_t$2 is the empirical distribution of $X_t$3 and $X_t$4 is obtained by deconvolving the distribution of $X_t$5.

The population-level identity behind the method is that if $X_t$6 is strictly increasing, then

$X_t$7

Thus monotonicity converts the unpaired problem into one of quantile transport. Theoretical validation takes two forms. First, if the estimation errors of $X_t$8 and $X_t$9 are bounded by $p(y_i\mid \lambda)$0 and $p(y_i\mid \lambda)$1, then $p(y_i\mid \lambda)$2 is sandwiched between nearby quantile-shifted values of $p(y_i\mid \lambda)$3. Second, under local Hölder conditions on $p(y_i\mid \lambda)$4 and $p(y_i\mid \lambda)$5,

$p(y_i\mid \lambda)$6

with probability at least $p(y_i\mid \lambda)$7. The method therefore complements matching: it can refine inference within matched strata, but it does not assume that matched records constitute genuine paired observations.

4. Sequential matching rules, permutation models, and discrepancy

The dependence induced by matching rules appears in especially explicit form in discrete coincidence problems. "On the Random Sampling of Pairs, with Pedestrian examples" compares two procedures for obtaining a matching pair from a discrete color distribution $p(y_i\mid \lambda)$8. Under the first procedure, two samples $p(y_i\mid \lambda)$9 are drawn independently and conditioned on matching, which yields

$\delta_i$0

Under the second procedure, sampling is sequential, with memory, until a color repeats; the distribution of the first repeated color is

$\delta_i$1

The discrepancy between the two induced pair-color laws is defined as the total variation distance

$\delta_i$2

The paper proves that $\delta_i$3 if and only if $\delta_i$4 is uniform, gives an exact maximizer for two colors with maximum discrepancy $\delta_i$5, an exact extremal configuration for three colors with discrepancy $\delta_i$6, and a candidate universal supremum $\delta_i$7 for the general discrete case (Arratia et al., 2012).

"A generalised matching distribution for the problem of coincidences" studies a related but more explicitly inferential setting. The classical matching distribution counts fixed points in a uniformly random permutation: $\delta_i$8 The generalization introduces a preliminary correct-matching stage: each of the $\delta_i$9 items is matched correctly with probability $y_i$0, and the remaining items are randomly permuted. If $y_i$1 denotes the number of pre-matched items and $y_i$2 the fixed-point count on the residual set, then

$y_i$3

This induces dependence because the second-stage permutation problem depends on the random residual size determined by the first stage. The resulting mass function is a convolution,

$y_i$4

with mean

$y_i$5

and variance

$y_i$6

For $y_i$7, the normalized count obeys a central limit theorem, and $y_i$8 converges in probability to $y_i$9. The paper develops exact and approximate tests of $\pi_i=P(\delta_i=1)$0 against $\pi_i=P(\delta_i=1)$1, thereby turning a dependent matching mechanism into a formal model for inference on matching ability (O'Neill, 2021).

5. Dependent missingness in matrix completion for matching markets

In two-sided matching markets, dependence appears as structured missingness. "Statistical Inference for Matching Decisions via Matrix Completion under Dependent Missingness" assumes a latent low-rank reward matrix

$\pi_i=P(\delta_i=1)$2

with singular value decomposition $\pi_i=P(\delta_i=1)$3 and incoherence conditions on $\pi_i=P(\delta_i=1)$4 and $\pi_i=P(\delta_i=1)$5. Across $\pi_i=P(\delta_i=1)$6 matching rounds, the observed data take the form

$\pi_i=P(\delta_i=1)$7

where $\pi_i=P(\delta_i=1)$8 is a matching matrix. The observed entries are not independent because $\pi_i=P(\delta_i=1)$9 is constrained by capacities: in one-to-one matching no two observed entries in a round can share a row or a column; in one-to-many matching a row can contain multiple ones but columns still cannot overlap; in two-sided random arrival the number of matches is itself random (Duan et al., 30 Oct 2025).

The paper analyzes three canonical mechanisms. In one-to-one matching, $0$0 is drawn uniformly from a constrained set and each entry has sampling probability $0$1. In one-to-many matching with one-sided random arrival, the per-entry sampling probability is $0$2, and the expected number of observed entries per round is $0$3. In two-sided random arrival, $0$4, where $0$5 and $0$6. The resulting observation process is therefore dependent even when the latent reward model is low rank.

Estimation is based on a non-convex factorization objective,

$0$7

optimized by Grassmannian gradient descent with sample splitting and rotation calibration. The estimator attains entrywise max-norm error bounds of the form

$0$8

for one-to-one matching,

$0$9

for one-to-many matching, and

$$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$0

for two-sided random arrival. The inference layer debiases the initial estimator and projects onto the low-rank tangent space to obtain asymptotic normality for arbitrary linear forms $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$1, including single entries, complete matchings, partial matchings, and differences between policies. The paper also gives plug-in variance estimators and confidence intervals adapted to each mechanism.

6. Conceptual distinctions, recurrent misconceptions, and limitations

Several distinctions recur across the literature. First, dependence induced by matching is not generally removed by marginal correction. In the survey-sampling setting, marginal weights $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$2 correct only first-order inclusion bias, whereas persistent within-household or within-pair dependence requires second-order weighting through $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$3 and control of higher-order inclusion probabilities (Williams et al., 2017). Second, matching on contextual covariates is not equivalent to observing genuine paired data. In the independent-database setting, the matched sample is induced by the matching rule, and the proposed estimator learns from the distributions within matched strata rather than from the constructed pairs themselves (Carpentier et al., 2016). Third, in coincidence problems, a “matching outcome” does not have a unique distribution unless the search rule is specified. The sock example shows that conditioning on a pairwise match and stopping at the first repeated color generate different laws, and the generalized matching distribution shows that correct pre-matching followed by random allocation is not a binomial model with independent indicators [(Arratia et al., 2012); (O'Neill, 2021)]. Fourth, in matching markets, observed entries cannot be treated as independently missing because row and column capacities couple observations within each round (Duan et al., 30 Oct 2025).

The principal limitations are model-specific and explicit. Pairwise pseudo-posterior inference requires access to second-order inclusion probabilities and assumptions on third- and fourth-order selection probabilities. Distributional reconstruction from independently collected samples requires monotonicity of $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$4, knowledge of the noise distribution $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$5 or a reasonable estimate of it, feasible deconvolution, and, for the main rate result, Hölder regularity. The generalized matching distribution parameterizes “matching ability” through $$\pi_{ij}=P(\delta_i=1\cap \delta_j=1), \qquad w_{ij}\propto \frac{1}{\pi_{ij}},$$6 and relies on a specific two-stage allocation mechanism. Matrix completion under dependent missingness assumes low rank, incoherence, adequate signal-to-noise ratio, and the validity of the prescribed matching mechanisms. A plausible implication is that matching-induced dependent sampling is best understood not as a single estimator class but as a family of problems in which the observational design and the inferential target must be aligned at the same structural level: pairs for pairwise survey inference, matched marginal strata for database integration, search rules for coincidence laws, and constrained matchings for matrix completion.